27 problems

The paper constructs a weak self-similar solution of the two-dimensional incompressible Euler equations for sufficiently small non-axisymmetric perturbations of the specified perio…

Šverák's conjecture. For a generic vorticity $\omega_0\in L^\infty(\mathbb{T}^2)$, the vorticity orbit

Let $(M,\mathsf{g})$ be a closed, compact Riemannian manifold without boundary, and let $(u,p)$ be a smooth solution of the incompressible Euler equations on $M$. Tao's conjecture.…

Let $u$ be a weak solution of the Euler equations whose spatial regularity is below $1/3$ of a derivative, or whose time integrability is below $q=\infty$, and let $e(t)$ denote it…

Sharpness conjecture. For every $\beta<\frac{1}{3}$, there exists a non-conservative weak solution

A vortex patch is a solution of the three-dimensional incompressible Euler equations whose initial vorticity has patch-like, conormally smooth structure; its boundary is initially…

A helical vortex tube is a vortex structure invariant under a helical symmetry, and a three-dimensional incompressible Euler flow is a velocity field satisfying the incompressible…

Elgindi's blow-up conjecture. For every $\alpha\in(0,1/3)$, there exists a compactly supported $C_c^{1,\alpha}$ axisymmetric swirl-free local solution to the Euler equations that d…

Rosenzweig's optimal scaling conjecture. This scaling assumption should be in general optimal: the incompressible Euler equation should not be the limiting evolution of the empiric…

Let $u$ be a smooth solution of the incompressible Euler equations in $4\mathbb{R}^3$ whose vorticity is initially concentrated near a smooth curve. Let the curve evolve according…

Let $T_\rho$ denote the time of validity of the evolution result for helical vortices with general initial data, and let $T^*$ be the corresponding limiting time in the vortex-ring…

Let $m$ be the number of branches in an $m$-fold symmetric algebraic spiral vortex sheet solution of the two-dimensional incompressible Euler equations, with…

Radial-profile conjecture. After a suitable scaling, the limit of the maximizers should be a radially decreasing function with respect to some point.

Uniqueness conjecture for maximizers. At least for some special rearrangement classes, there exists $\hat v\in\mathcal M$ such that

Local dissipativity conjecture. The limit probability measures of the theorem are supported not only on weak Euler solutions but in fact on solutions which are locally dissipative.

Rosenzweig's optimality conjecture. The scaling assumption above should be optimal in general: the incompressible Euler equation should not be the limiting evolution of the empiric…

Let $D$ be an open disk centered at the origin. Let $z\in D$ and let $\mathbf{v}\in C^2(\overline{D}\backslash \\{z\\})$ be a bounded flow solving the Euler equations on…

Uniqueness extension conjecture. The uniqueness theorem for these steady vortex rings should also hold when

$5/6$ threshold conjecture. For every $s<\frac{5}{6}$, there are weak solutions to the Euler equation in $C_tH^s_x$ with strictly decreasing kinetic energy.

Let $u$ be a weak solution of the incompressible Euler equations that is sufficiently rough and dissipative, and let the behavior denoted by … . This conjecture is the basis for in…

A vortex sheet is a solution of the two-dimensional incompressible Euler equation whose velocity field is discontinuous across an interface; here the initial interface data are ass…

Let $\Omega$ be a multiply connected fluid domain with boundary decomposed into entering, exiting, and impermeable parts $\Gamma_{\mathrm{in}}$, $\Gamma_{\mathrm{out}}$, and…

Isett–Oh conjecture. For every $\theta<\frac{1}{3}$, there exists a weak Euler solution $v\in C^\theta(\mathbb R\times\mathbb T^n)$ such that

Finite-time blowup conjecture. There exists such a compact Riemannian manifold $(M,g)$, of some dimension $d>2$, and such a smooth solution that cannot be smoothly continued to the…

Let $\omega_0\in H^1(\mathbb{T}^2)\cap L^\infty(\mathbb{T}^2)$ be mean-zero, and let the corresponding solution of the two-dimensional incompressible Euler equations be a Yudovich…